Could the "post-forcing continuum" be a new cardinality?

This question is vaguely related to this MO question of mine.

Suppose $V$ is a (for simplicity, well-founded) model of $mathsf{ZF+AD}$. Let $mathbb{P}$ be a forcing notion in $V$, and let $G$ be $mathbb{P}$-generic over $V$. My question is whether a certain "new cardinality" phenomenon can occur:

Could it be the case that $V[G]models$ "There is no bijection between $mathbb{R}^{V[G]}$ and any $xin V$"?

One wrinkle here is that $V[G]$ need not satisfy $mathsf{AD}$ anymore; indeed, preserving $mathsf{AD}$ is quite hard. So $mathsf{AD}$ doesn’t give us a lot of "leverage" in the forcing extension that I can see.

I would be happy to replace $mathsf{ZF+AD}$ with any stronger (consistent :P) theory; in particular, it might be easier to get a negative answer for something more "canonical" like $mathsf{ZF+AD}+V=L(mathbb{R})$.

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Here are a couple quick comments:

On the one hand, to preempt worries about triviality note that we can have $V[G]models$ "There is some $a$ which is not in bijection with any $bin V$." For example, Monro showed that we can have an amorphous set in $V[G]$ even if there are no amorphous sets in $V$; since amorphousness is downwards-absolute, any amorphous set in such a $V[G]$ is not (in $V[G]$, anyways) in bijection with any set in $V$. (This reference was pointed out to me by Asaf Karagila.)

On the other hand, questions of this type are always trivial over $mathsf{ZFC}$ for the following reason:

• Forcing preserves choice, so $V[G]modelsmathsf{ZFC}$.

• That means that for every name $nu$ we have in $V[G]$ that $nu[G]$ is in bijection with some ordinal $alphain Ord^{V[G]}$.

• But $Ord^V=Ord^{V[G]}$, so $V[G]$ sees a bijection between $nu[G]$ and some set in $V$.

On the other other hand, the previous hand’s argument is quite fragile and I don’t see that it gives any insight into my question. Specifically, the third bulletpoint is crucial but has no relevant analogue for the $mathsf{ZF+AD}$-setting that I can see.